Let/. It :0 ---0 G be any two self maps of a compact connected oriented Lie group G. In this paper, for each positive integer k , we associate an integer with fk,hi . We relate this number with Lefschetz coincidence number. We deduce that for any two differentiable maps f, there exists a positive integer k such that k 5.2+1 , and there is a point x C G such that ft (x) = (x) , where A is the rank of G . Introduction Let G be an n-dimensional com -pact connected Lie group with multip-lication p ( .e 44:0 xG--+G such that p ( x , y) = x.y ) and unit e . Let [G, G] be the set of homotopy classes of maps G G . Given two maps f , f G ---• Jollowing [3], we write f. f 'to denote the map G-.Gdefined by 01.11® =A/WO= fiat® ,sea Given a point g EC and a differ-entiable map F: G G , write GA to denote the tangent space of G at g [4,p.10] , and denote by d x F the linear map rig F :Tx0 T, (x)G induced by F , it is called the differential of Fat g [4,p.22]. Let LA, Rx :0 G be respec-tively the left translation Lx(i)=4..(g,e) , and the right translation Rx(1)./..(gcg). Then there is a natural homomorphism Ad ,the adjoin representation, from G to GL(G•), (the group of nonsingular linear transformations of Qdefined as follows:- Ad(g)= deRe, od,Lx. Note that d xRc, ad.; =d(4,( Lx(e)))0 de; =d.(4, 04)=4(40 Re) = d(4(4, (e)))0 (44, =d ar, o (44, . Since G is connected , the image of Ad belongs to the connected component of G(G)containing the identity,i.e. for each g E 0, detAd(g) > 0 . By Exercise Al • Dr.-Prof.-Department of Mathematics- College of Science- University of Baghdad. •• Dr.-Department of Mathematics- College of Science for Woman- University of Baghdad.
Non-alcoholic fatty liver disease (NAFLD) is one of chronic liver and defines by fat accumulation ≥5% in liver which can progresses to non-alcoholic steatohepatitis (NASH). NAFLD related to obesity as well as non obese individuals. Adiponectin is a cytokine secreted from adipose tissue involved NAFLD pathogenesis and liked with obesity. Irisin is a myokine, has a convenient effect against metabolic diseases such as obesity, disylipemia diabetes type 2 and reversed liver steatosis and may be related with NAFLD.
Vitamin D is one of the fat soluble vitamins and more precisely as a pro-hormone through its metabolite (1,25(OH)2 cholecalciferol) the major steroid hormone. After the skin exposure to the light, vitamin D un
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Release of industrial effluents comprising dyes in water bodies is one of the foremost causes of water pollution. Therefore, the proper and proficient treatment of these dyes contaminated left-over material before their release is crucial. Herein, an eco-friendly biological macromolecule Gum-Acacia (GA) integrated Fe3O4 nanoparticles composite hydrogel was manufactured via co-precipitation technique for effective adsorption of Congo red (CR) dye existing in water bodies. The as-prepared magnetic GA/Fe3O4 composite hydrogel was characterized by FTIR, XRD, EDX, VSM, SEM, and BET techniques. These studies discovered the fruitful fabrication of biodegradable magnetic GA/Fe3O4 composite hydrogel possessing porous structure with large surface are
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Water samples from a variety of sources in Kelantan, Malaysia (lakes, ponds, rivers, ditches, fish farms, and sewage) were screened for the presence of bacteriophages infecting
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Oral squamous cell carcinoma (OSCC) is the most common malignant neoplasm of the oral mucosa. Human papillomavirus (HPV) virus cause a broad scope of diseases from benign to invasive tumors, types 16 and 18 classified as carcinogenic to humans. This study aimed to provide the first molecular characterization of HPV types in Iraq. Thirty-five unstimulated whole saliva samples were collected from histopathologically confirmed patients with oral cancer were enrolled in this study. Genomic DNA was extracted from exfoliating cells to amplify HPV-DNA using HPV-L1 gene sequence primers by polymerase chain reaction method (PCR), the viral genotyping was performed using direct sequencing method. HPV genotypes identified were deposited in Gen
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Software-defined networks (SDN) have a centralized control architecture that makes them a tempting target for cyber attackers. One of the major threats is distributed denial of service (DDoS) attacks. It aims to exhaust network resources to make its services unavailable to legitimate users. DDoS attack detection based on machine learning algorithms is considered one of the most used techniques in SDN security. In this paper, four machine learning techniques (Random Forest, K-nearest neighbors, Naive Bayes, and Logistic Regression) have been tested to detect DDoS attacks. Also, a mitigation technique has been used to eliminate the attack effect on SDN. RF and KNN were selected because of their high accuracy results. Three types of ne
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Non-alcoholic fatty liver disease (NAFLD) is one of chronic liver and defines by fat accumulation ≥5% in liver which can progresses to non-alcoholic steatohepatitis (NASH). NAFLD related to obesity as well as non obese individuals. Adiponectin is a cytokine secreted from adipose tissue involved NAFLD pathogenesis and liked with obesity. Irisin is a myokine, has a convenient effect against metabolic diseases such as obesity, disylipemia diabetes type 2 and reversed liver steatosis and may be related with NAFLD. Vitamin D is one of the fat soluble vitamins and more precisely as a pro-hormone through its metabolite (1,25(OH)2 cholecalciferol) the major steroid hormone. After the skin exposure to the light, vitamin D undergoes to
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Previously many properties of graphene oxide in the field of medicine, biological environment and in the field of energy have been studied. This diversity in properties is due to the possibility of modification on the composition of this Nano compound, where the Graphene oxide is capable of more modification via addition other functional groups on its surface or at the edges of the sheet. The reason for this modification possibility is that the Sp3 hybridization (tetrahedral structure) of the carbon atoms in graphene oxide, and it contains many oxygenic functional groups that are able to reac with other groups. In this research the effect of addition of some amine compounds on electrical properties of graphene oxide has been studied by the
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The drive of this exploration is to investigate the mucoadhesive assets of A. indica (Azadirachta indica) fruit mucilage by incorporating it into mucoadhesive microspheres with Acyclovir (AVR) as a model drug. The study was performed to check the impact of the mucilage proportion on particle size and swelling index. Nine batches of AVR mucoadhesive microspheres were made with varying proportions of Polyacrylic acid 934P and A. indica fruit mucilage (AIFM). A central composite design with design expert software to check the impact of dependent variables (A. indica mucilage and Polyacrylic acid 934 P levels) on particle size and swelling index as a response. As part of congeniality studies, the batches w
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